Fast and Simple 2D Geometric Proximity Queries Using Graphics Hardware

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• Fast and Simple 2D Geometric Proximity Queries
  Using Graphics Hardware
• Kenneth E. Hoff III, Andrew Zaferakis,
• Ming Lin, and Dinesh Manocha
• University of North Carolina at Chapel Hill
• I3D 2001

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Motivation
Colliding?
Yes / No
Contact Pairs
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Motivation
Colliding?
Yes / No
Contact Pairs
More information is required for computing
collision responses!
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Motivation
Colliding?
Exact
Yes / No
Contact Pairs
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Motivation
Colliding?
Exact
Yes / No
Contact Pairs
Approximate
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Proximity Queries
Intersections
Contact Points Normals
Closest Points,Separation Distance
Penetration Depth
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Our Approach
Hybrid Approach
Object-space Localization
Image-SpaceProximity Query
ProximityInformation
CPU
Graphics-HardwareAcceleration
Bounded-ErrorApproximation
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Related Work

• Proximity query algorithms
• Exact, object-space
• Approximate, image-space
• Distance fields

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Exact, Object-Space Algorithms
Preprocessing
Static Rigid Objects
Convex decomposition
Hierarchical data structures
Gilbert88 , Lin90, Cameron97, Ehmann00
Gottschalk96, Hubbard93, Johnson98, Quinlan94
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Exact, Object-Space Algorithms
Restricted Cases

• Dynamic
• Deformable
• Objects

Known surfacemotion trajectories
Specialized geometry (cloth)
Snyder93
Baraff92, Baraff98
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Approximate, Image-Space Algorithms

• Algorithms simplified and accelerated using
  graphics hardware
• Visibility Sutherland74, Catmull75
• Intersections Rossignac92, Myskowski95, Baciu98
• CSG Goldfeather86, Rossignac90, Stewart00
• Robot motion-planning Lengyel90, Hoff00
• Voronoi diagrams Hoff99
• Advantages
• Simplicity
• Bounded error
• Linear complexity
• Robustness

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Distance Fields

• Distance fields give distance to nearest object
  from any point
• Fast marching method Sethian96
• Generalized Voronoi diagrams Hoff99
• Adaptively-sampled distance fields Frisken00
• Distance fields for penetration
  computation Fisher01

Distance field
3 point objects
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Distance Fields

• Voronoi diagram computation using graphics
  hardware (Hoff99)

Render polygonal mesh approximations of object
distance fields
Color buffer
Depth buffer Result after compositing distance
fields using minimum depth test
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Our Goal

• Proximity query algorithm with the following
  properties
• Performs all proximity queries
• Handles non-convex primitives
• Requires no precomputation or complex data
  structures
• Fast and efficient
• Robust
• Portable
• Bounded error
• No previous algorithm found, even in 2D!

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Our Hybrid Approach
CPU
Graphics
Image-space proximity queries
Coarse object-spacegeometric localization
Balance load by varying localization coarseness
and error bound
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Geometric Localization
Potential Closest Points
Potential Intersections

• Reduces the number of pixels processed per
  image-space query
• Vary max depth in hierarchy to balance
  CPU/graphics loads

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Image-Space Proximity Queries

• Intersections
• Distance field and gradients

• Contact points and normals
• Closest points
• Separation distance and direction
• Penetration depth and direction

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Intersections

• Boundary-boundary

Boundary-volume
Volume-volume
B
A
?
?
Draw A
Draw B
Intersection Overwritten Pixels
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Distance Field and Gradients
Draw polygon to encode negative sign in a buffer
Central difference to compute gradient at a pixel
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Image-Space Proximity Queries

• Intersections
• Distance field and gradients

• Contact points and normals
• Closest points
• Separation distance and direction
• Penetration depth and direction

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Contact Points and Normals
Normals from gradient of distance field
Close contact points as intersection of enlarged
boundary
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Closest Points
Find closest point on object A to object B
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Closest Points
Localize region on A containing the closest point
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Closest Points
Compute distance field of B in As localized
region
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Closest Points

• Render boundary of A to find set of potential
  closest points
• Check distances at each point to find the closest

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Separation Distance and Direction
Distance at closest point min separation
distanceGradient at closest point separating
axis direction
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Penetration Depth
Minimum translational distance needed to separate
objects
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Penetration Depth and Direction for a Point
B
A
d
Compute penetration depth anddirection for As
boundary pointsthat intersect Bs interior
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Putting it all together
Collision response between 2 deformable objects
A
B
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Localize Potential Intersections
A
B
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Find Penetrating Points
A
B
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Tighten Localized Region
A
B
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Compute Distance Field
A
B
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Compute Penetration Depth and Direction
A
B
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Repeat Process for Other Object
A
B
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Use Proximity Info for Collision Response
A
B
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Separation Forces from Penetration Depth
A
B
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Demonstrations

• Simulation test cases
• Rigid and deformable objects
• Different contact scenarios
• Collision response in 2 simulation strategies
• Non-penetration constraint, backtracking
• Unconstrained, penalty-based
• In each demo
• No precomputation
• Interactive frame rates
• Bounding-box intersection localization
• Pentium III 800, nVidia GeForce256

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MAP

• Non-convex, rigid objects
• Frequent simultaneous close contacts
• Unconstrained, penalty-based

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BUMPER CARS

• Convex, rigid objects
• Less frequent contact
• Non-penetration constraint, backtracking

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LINKS

• Non-convex, rigid objects
• Frequent interlocking contacts
• Unconstrained, penalty-based

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GEARS

• Non-convex, rigid objects
• Less frequent interlocking contacts
• Unconstrained, penalty-based

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WAVY

• Non-convex, deformable objects
• Continuous contact
• Unconstrained, penalty-based

Resolving extreme penetration
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Conclusion

• Hybrid object/image space proximity query
  algorithm using graphics hardware acceleration
• Desired goals have been met for 2D objects

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Summary of Properties

• Simple to implement
• Useful in a wide range of applications
• Running time related to the rendering time
• Error bound provides smooth dial between
  performance and level of approximation
• Varying the max depth in the hierarchical
  localization allows balancing load between CPU
  and graphics

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Current Work Extension to 3D
3D distance field
3D intersections
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Acknowledgements

• Sarah Hoff
• UNC Computer Science Dept.
• ARO, NSF, NIH, ONR, Intel

http//www.cs.unc.edu/geom/PIVOT/ Releasing the
libraries very soon! Demonstration tonight at UNC
open house
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Performance
Performance timings for dynamics simulations. The
number of objects, number of line segments, and
the average total time in milliseconds to run
proximity queries on all objects in the scene per
frame is reported. Timing data was gathered from
three machines a Pentium-III 933MHz desktop with
a 64Mb GeForce2, a SGI 300MHz R12000 with
InfiniteReality2 graphics, and a Pentium-III
750Mhz laptop with ATI Rage Pro LT graphics.
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Effects of Changes in Error Tolerance
The effect on performance when changing the
distance error tolerance d. We used proximity
queries on the wavy demo with no collision
response. The error determines the number of
pixels used in the image-based operations.
Systems with low graphics performance are more
directly affected by the choice of d (see ATI
Rage Pro LT) however, as the error is increased
there is less dependence on graphics performance
and the faster laptop CPU overtakes the
InfiniteReality2 system.